Solve 1/6(4x+5)-2/3(2x+7)=3

$$\frac{ 1 }{ 6 } (4x+5) \frac{ -2 }{ 3 } (2x+7)=3$$

Solve for x (complex solution)

$$x=\frac{-19+3\sqrt{15}i}{8}\approx -2.375+1.452368755i$$

$$x=\frac{-3\sqrt{15}i-19}{8}\approx -2.375-1.452368755i$$

Steps Using the Quadratic Formula

Fraction \frac{-2}{3} can be rewritten as -\frac{2}{3} by extracting the negative sign.

$$\frac{1}{6}\left(4x+5\right)\left(-\frac{2}{3}\right)\left(2x+7\right)=3$$

Multiply \frac{1}{6} and -\frac{2}{3} to get -\frac{1}{9}.

$$-\frac{1}{9}\left(4x+5\right)\left(2x+7\right)=3$$

Use the distributive property to multiply -\frac{1}{9} by 4x+5.

$$\left(-\frac{4}{9}x-\frac{5}{9}\right)\left(2x+7\right)=3$$

Use the distributive property to multiply -\frac{4}{9}x-\frac{5}{9} by 2x+7 and combine like terms.

$$-\frac{8}{9}x^{2}-\frac{38}{9}x-\frac{35}{9}=3$$

Subtract 3 from both sides.

$$-\frac{8}{9}x^{2}-\frac{38}{9}x-\frac{35}{9}-3=0$$

Subtract 3 from -\frac{35}{9} to get -\frac{62}{9}.

$$-\frac{8}{9}x^{2}-\frac{38}{9}x-\frac{62}{9}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{8}{9} for a, -\frac{38}{9} for b, and -\frac{62}{9} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{38}{9}\right)±\sqrt{\left(-\frac{38}{9}\right)^{2}-4\left(-\frac{8}{9}\right)\left(-\frac{62}{9}\right)}}{2\left(-\frac{8}{9}\right)}$$

Square -\frac{38}{9} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-\frac{38}{9}\right)±\sqrt{\frac{1444}{81}-4\left(-\frac{8}{9}\right)\left(-\frac{62}{9}\right)}}{2\left(-\frac{8}{9}\right)}$$

Multiply -4 times -\frac{8}{9}.

$$x=\frac{-\left(-\frac{38}{9}\right)±\sqrt{\frac{1444}{81}+\frac{32}{9}\left(-\frac{62}{9}\right)}}{2\left(-\frac{8}{9}\right)}$$

Multiply \frac{32}{9} times -\frac{62}{9} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{38}{9}\right)±\sqrt{\frac{1444-1984}{81}}}{2\left(-\frac{8}{9}\right)}$$

Add \frac{1444}{81} to -\frac{1984}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{38}{9}\right)±\sqrt{-\frac{20}{3}}}{2\left(-\frac{8}{9}\right)}$$

Take the square root of -\frac{20}{3}.

$$x=\frac{-\left(-\frac{38}{9}\right)±\frac{2\sqrt{15}i}{3}}{2\left(-\frac{8}{9}\right)}$$

The opposite of -\frac{38}{9} is \frac{38}{9}.

$$x=\frac{\frac{38}{9}±\frac{2\sqrt{15}i}{3}}{2\left(-\frac{8}{9}\right)}$$

Multiply 2 times -\frac{8}{9}.

$$x=\frac{\frac{38}{9}±\frac{2\sqrt{15}i}{3}}{-\frac{16}{9}}$$

Now solve the equation x=\frac{\frac{38}{9}±\frac{2\sqrt{15}i}{3}}{-\frac{16}{9}} when ± is plus. Add \frac{38}{9} to \frac{2i\sqrt{15}}{3}.

$$x=\frac{\frac{2\sqrt{15}i}{3}+\frac{38}{9}}{-\frac{16}{9}}$$

Divide \frac{38}{9}+\frac{2i\sqrt{15}}{3} by -\frac{16}{9} by multiplying \frac{38}{9}+\frac{2i\sqrt{15}}{3} by the reciprocal of -\frac{16}{9}.

$$x=\frac{-3\sqrt{15}i-19}{8}$$

Now solve the equation x=\frac{\frac{38}{9}±\frac{2\sqrt{15}i}{3}}{-\frac{16}{9}} when ± is minus. Subtract \frac{2i\sqrt{15}}{3} from \frac{38}{9}.

$$x=\frac{-\frac{2\sqrt{15}i}{3}+\frac{38}{9}}{-\frac{16}{9}}$$

Divide \frac{38}{9}-\frac{2i\sqrt{15}}{3} by -\frac{16}{9} by multiplying \frac{38}{9}-\frac{2i\sqrt{15}}{3} by the reciprocal of -\frac{16}{9}.

$$x=\frac{-19+3\sqrt{15}i}{8}$$

The equation is now solved.

$$x=\frac{-3\sqrt{15}i-19}{8} x=\frac{-19+3\sqrt{15}i}{8}$$

Steps for Completing the Square

Fraction \frac{-2}{3} can be rewritten as -\frac{2}{3} by extracting the negative sign.

$$\frac{1}{6}\left(4x+5\right)\left(-\frac{2}{3}\right)\left(2x+7\right)=3$$

Multiply \frac{1}{6} and -\frac{2}{3} to get -\frac{1}{9}.

$$-\frac{1}{9}\left(4x+5\right)\left(2x+7\right)=3$$

Use the distributive property to multiply -\frac{1}{9} by 4x+5.

$$\left(-\frac{4}{9}x-\frac{5}{9}\right)\left(2x+7\right)=3$$

Use the distributive property to multiply -\frac{4}{9}x-\frac{5}{9} by 2x+7 and combine like terms.

$$-\frac{8}{9}x^{2}-\frac{38}{9}x-\frac{35}{9}=3$$

Add \frac{35}{9} to both sides.

$$-\frac{8}{9}x^{2}-\frac{38}{9}x=3+\frac{35}{9}$$

Add 3 and \frac{35}{9} to get \frac{62}{9}.

$$-\frac{8}{9}x^{2}-\frac{38}{9}x=\frac{62}{9}$$

Divide both sides of the equation by -\frac{8}{9}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-\frac{8}{9}x^{2}-\frac{38}{9}x}{-\frac{8}{9}}=\frac{\frac{62}{9}}{-\frac{8}{9}}$$

Dividing by -\frac{8}{9} undoes the multiplication by -\frac{8}{9}.

$$x^{2}+\left(-\frac{\frac{38}{9}}{-\frac{8}{9}}\right)x=\frac{\frac{62}{9}}{-\frac{8}{9}}$$

Divide -\frac{38}{9} by -\frac{8}{9} by multiplying -\frac{38}{9} by the reciprocal of -\frac{8}{9}.

$$x^{2}+\frac{19}{4}x=\frac{\frac{62}{9}}{-\frac{8}{9}}$$

Divide \frac{62}{9} by -\frac{8}{9} by multiplying \frac{62}{9} by the reciprocal of -\frac{8}{9}.

$$x^{2}+\frac{19}{4}x=-\frac{31}{4}$$

Divide \frac{19}{4}, the coefficient of the x term, by 2 to get \frac{19}{8}. Then add the square of \frac{19}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{19}{4}x+\left(\frac{19}{8}\right)^{2}=-\frac{31}{4}+\left(\frac{19}{8}\right)^{2}$$

Square \frac{19}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{19}{4}x+\frac{361}{64}=-\frac{31}{4}+\frac{361}{64}$$

Add -\frac{31}{4} to \frac{361}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{19}{4}x+\frac{361}{64}=-\frac{135}{64}$$

Factor x^{2}+\frac{19}{4}x+\frac{361}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{19}{8}\right)^{2}=-\frac{135}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{19}{8}\right)^{2}}=\sqrt{-\frac{135}{64}}$$

Simplify.

$$x+\frac{19}{8}=\frac{3\sqrt{15}i}{8} x+\frac{19}{8}=-\frac{3\sqrt{15}i}{8}$$

Subtract \frac{19}{8} from both sides of the equation.

$$x=\frac{-19+3\sqrt{15}i}{8} x=\frac{-3\sqrt{15}i-19}{8}$$